4.3 Automatic Dimensioning
57
For modules, its geometry specification and the flow rate allow to determine the
required time, i.e. by
l m w m h m
Q m
≤ T or
l m w m h m
Q m
≥ T
(4.6)
Accordingly, if an upper/lower bound for a sequence of channels/modules is
needed, the respective times of the single components can be summed up and,
afterwards, restricted.
Example 4.7 Let’s assume the same objectives as specified in Examples 4.3 and 4.4
are applied. In order to ensure these, the method adds for all channels and modules
the inequalities
Q c ≥ 0 μl/min and Q m ≥ 0 μl/min.
(4.7)
The upper time limit T = 500 ms is ensured by
c∈{c 5 ,c 6 ,c 10 }
R c w 2
c h 4
c
α μ Q c
≤ 500 ms.
(4.8)
Finally, the ranges of the Reynolds number and the Capillary number can be
restricted. Therefore, the method adds for all channels and modules the inequalities
ρ Q c L
μ cont w c h c
≤ 1 and
ρ Q m L
μ cont w m h m
≤ 1
(4.9)
for the Reynolds number as well as
μ cont Q c
γ w c h c
< 10
−2 and
μ cont Q m
γ w m h m
< 10
−2
(4.10)
for the Capillary number.
Successfully solving the resulting equation system yields values for all
variables—including values for the resistances and flow rates of each channel. From
this, the respective dimensions of the channels can be derived and the specification
is completed. In contrast, if it can be proven that no solution for this equation system
exists, it has been shown that the partial specification does not allow to fulfill all
objectives.
Example 4.8 Recall that the designer initially dimensioned all remaining channels
with the same resistance of 0.15 mbar/(μl/min) (cf. Example 4.2). Solving the
equation system resulting from the steps above yields slightly different resistances,
namely (given in mbar/(μl/min)):
57
For modules, its geometry specification and the flow rate allow to determine the
required time, i.e. by
l m w m h m
Q m
≤ T or
l m w m h m
Q m
≥ T
(4.6)
Accordingly, if an upper/lower bound for a sequence of channels/modules is
needed, the respective times of the single components can be summed up and,
afterwards, restricted.
Example 4.7 Let’s assume the same objectives as specified in Examples 4.3 and 4.4
are applied. In order to ensure these, the method adds for all channels and modules
the inequalities
Q c ≥ 0 μl/min and Q m ≥ 0 μl/min.
(4.7)
The upper time limit T = 500 ms is ensured by
c∈{c 5 ,c 6 ,c 10 }
R c w 2
c h 4
c
α μ Q c
≤ 500 ms.
(4.8)
Finally, the ranges of the Reynolds number and the Capillary number can be
restricted. Therefore, the method adds for all channels and modules the inequalities
ρ Q c L
μ cont w c h c
≤ 1 and
ρ Q m L
μ cont w m h m
≤ 1
(4.9)
for the Reynolds number as well as
μ cont Q c
γ w c h c
< 10
−2 and
μ cont Q m
γ w m h m
< 10
−2
(4.10)
for the Capillary number.
Successfully solving the resulting equation system yields values for all
variables—including values for the resistances and flow rates of each channel. From
this, the respective dimensions of the channels can be derived and the specification
is completed. In contrast, if it can be proven that no solution for this equation system
exists, it has been shown that the partial specification does not allow to fulfill all
objectives.
Example 4.8 Recall that the designer initially dimensioned all remaining channels
with the same resistance of 0.15 mbar/(μl/min) (cf. Example 4.2). Solving the
equation system resulting from the steps above yields slightly different resistances,
namely (given in mbar/(μl/min)):